Problem: $f(x, y) = x^3y$ What is $\dfrac{\partial f}{\partial x}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $3x^2y + x^3$ (Choice B) B $x^3$ (Choice C) C $3x^2y$ (Choice D) D $3x^2$
Solution: Taking a partial derivative with respect to $x$ means treating $y$ like a constant, then taking a normal derivative. $\begin{aligned} \dfrac{\partial f}{\partial x} &= \dfrac{\partial}{\partial x} \left[ {x^3} y \right] \\ \\ &= {3x^2} y \end{aligned}$ In conclusion, $\dfrac{\partial f}{\partial x} = 3x^2 y$